The microstructure of a hot-work tool steel additively manufactured using laser powder-bed fusion (L-PBF), and its response to post heat treatment, is studied in detail by microstructure characterization and computational thermodynamics and kinetics. The high solidification and cooling rates during the L-PBF process lead to suppression of δ-ferrite and instead solidification of an austenite phase directly containing a cellular substructure where the alloying elements have segregated to the inter-cellular regions and where solidification carbides have formed in the cell junctions. The austenite is then partly decomposed into martensite at lower temperatures. The micro-segregation can be predicted by reducing the complex solidification behavior to a diffusion problem in one dimension enabling detailed comparisons with the measured segregation profiles quantified at a nanometer scale. Martensite start temperature (Ms) calculations along the spatially varying composition show that the Ms temperature decreases in the inter-cellular regions where austenite is observed. The network of austenite in the as-built microstructure can be understood from the combined influence of the composition dependence of the Ms temperature in relation to the build plate temperature and the mechanical stabilization of the small-sized austenite regions. This work demonstrates the power of computational tools based on computational thermodynamics and kinetics for designing tool steels for additive manufacturing by predictions of the steel's response to the L-PBF process and post heat treatments.